Elliptic Curve Random Number Generation

ABSTRACT

An elliptic curve random number generator avoids escrow keys by choosing a point Q on the elliptic curve as verifiably random. An arbitrary string is chosen and a hash of that string computed. The hash is then converted to a field element of the desired field, the field element regarded as the x-coordinate of a point Q on the elliptic curve and the x-coordinate is tested for validity on the desired elliptic curve. If valid, the x-coordinate is decompressed to the point Q, wherein the choice of which is the two points is also derived from the hash value. Intentional use of escrow keys can provide for back up functionality. The relationship between P and Q is used as an escrow key and stored by for a security domain. The administrator logs the output of the generator to reconstruct the random number with the escrow key.

This application claims priority from U.S. Provisional PatentApplication No. 60/644,982 filed on Jan. 21, 2005.

FIELD OF THE INVENTION

The present invention relates to systems and methods for cryptographicrandom number generation.

DESCRIPTION OF THE PRIOR ART

Random numbers are utilised in many cryptographic operations to provideunderlying security. In public key infrastructures, for example, theprivate key of a key pair is generated by a random number generator andthe corresponding public key mathematically derived therefrom. A new keypair may be generated for each session and the randomness of thegenerator therefore is critical to the security of the cryptographicsystem.

To provide a secure source of random numbers, cryptographically securepseudorandom bit generators have been developed in which the security ofeach generator relies on a presumed intractability of the underlyingnumber-theoretical problem. The American National Standards Institute(ANSI) has set up an Accredited Standards Committee (ASC) X9 for thefinancial services industry, which is preparing a American NationalStandard (ANS) X9.82 for cryptographic random number generation (RNG).One of the RNG methods in the draft of X9.82, called Dual_EC_DRBG, useselliptic curve cryptography (ECC) for its security. Dual_EC_DRBG willhereinafter be referred to as elliptic curve random number generation(ECRNG).

Elliptic curve cryptography relies on the intractability of the discretelog problem in cyclic subgroups of elliptic curve groups. An ellipticcurve E is the set of points (x, y) that satisfy the defining equationof the elliptic curve. The defining equation is a cubic equation, and isnon-singular. The coordinates x and y are elements of a field, which isa set of elements that can be added, subtracted and divided, with theexception of zero. Examples of fields include rational numbers and realnumbers. There are also finite fields, which are the fields most oftenused in cryptography. An example of a finite field is the set ofintegers modulo a prime q.

Without the loss of generality, the defining equation of the ellipticcurve can be in the Weierstrass form, which depends on the field of thecoordinates. When the field F is integers modulo a prime q>3, then theWeierstrass equation takes the form y³=x³+ax+b, where a and b areelements of the field F.

The elliptic curve E includes the points (x, y) and one further point,namely the point O at infinity. The elliptic curve E also has a groupstructure, which means that the two points P and Q on the curve can beadded to form a third point P+Q. The point O is the identity of thegroup, meaning P+O=O+P=P, for all points P. Addition is associative, sothat P+(Q+R)=(P+Q)+R, and commutative, so that P+Q=Q+R, for all pointsP, Q and R. Each point P has a negative point −P, such that P+(−P)=O.When the curve equation is the Weierstrass equation of the formy²=x³+ax+b, the negative of P=(x,y) is determined easily as −P=(x, −y).The formula for adding points P and Q in terms of their coordinates isonly moderately complicated involving just a handful of fieldoperations.

The ECRNG uses as input two elliptic curve points P and Q that arefixed. These points are not assumed to be secret. Typically, P is thestandard generator of the elliptic curve domain parameters, and Q issome other point. In addition a secret seed is inserted into the ECRNG.

The ECRNG has a state, which may be considered to be an integers. Thestate s is updated every time the ECRNG produces an output. The updatedstate is computed as u=z(sP), where z( ) is a function that converts anelliptic curve point to an integer. Generally, z consists of taking thex-coordinate of the point, and then converting the resulting fieldelement to an integer. Thus u will typically be an integer derived fromthe x-coordinate of the point s.

The output of the ECRNG is computed as follows: r=t(z(sQ)), where is atruncation function. Generally the truncation function removes theleftmost bits of its input. In the ECRNG, the number of bits truncateddepends on the choice of elliptic curve, and typically may be in therange of 6 to 19 bits.

Although P and Q are known, it is believed that the output r is randomand cannot be predicted. Therefore successive values will have norelationship that can be exploited to obtain private keys and break thecryptographic functions. The applicant has recognised that anybody whoknows an integer d such that Q=dP, can deduce an integer e such thated=1 mod n, where n is the order of G, and thereby have an integer esuch that P=eQ. Suppose U=sP and R=sQ, which are the precursors to theupdated state and the ECRNG output. With the integer e, one can computeU from R as U=eR. Therefore, the output r=t(z(R)), and possible valuesof R can be determined from r. The truncation function means that thetruncated bits of R would have to be guessed. The z function means thatonly the x-coordinate is available, so that decompression would have tobe applied to obtain the full point R. In the case of the ECRNG, therewould be somewhere between about 2⁶=64 and 2¹⁹ (i.e. about half amillion) possible points R which correspond to r, with the exact numberdepending on the curve and the specific value of r.

The full set of R values is easy to determine from r, and as notedabove, determination of the correct value for R determines U=eR, if oneknows e. The updated state is u=z(U), so it can be determined from thecorrect value of R. Therefore knowledge of r and e allows one todetermine the next state to within a number of possibilities somewherebetween 2⁶ and 2¹⁹. This uncertainty will invariably be eliminated onceanother output is observed, whether directly or indirectly through aone-way function.

Once the next state is determined, all future states of ECRNG can bedetermined because the ECRNG is a deterministic function. (at leastunless additional random entropy is fed into the ECRNG state) Alloutputs of the ECRNG are determined from the determined states of theECRNG. Therefore knowledge of r and e, allows one to determine allfuture outputs of the ECRNG.

It has therefore been identified by the applicant that this methodpotentially possesses a trapdoor, whereby standardizers or implementersof the algorithm may possess a piece of information with which they canuse a single output and an instantiation of the RNG to determine allfuture states and output of the RNG, thereby completely compromising itssecurity. It is therefore an object of the present invention to obviateor mitigate the above mentioned disadvantages.

SUMMARY OF THE INVENTION

In one aspect, the present invention provides a method for computing averifiably random point Q for use with another point P in an ellipticcurve random number generator comprising computing a hash including thepoint P as an input, and deriving the point Q from the hash.

In another aspect, the present invention provides a method for producingan elliptic curve random number comprising generating an output using anelliptic curve random number generator, and truncating the output togenerate the random number.

In yet another aspect, the present invention provides a method forproducing an elliptic curve random number comprising generating anoutput using an elliptic curve random number generator, and applying theoutput to a one-way function to generate the random number.

In yet another aspect, the present invention provides a method of backupfunctionality for an elliptic curve random number generator, the methodcomprising the steps of computing an escrow key e upon determination ofa point Q of the elliptic curve, whereby P=eQ, P being another point ofthe elliptic curve; instituting an administrator, and having theadministrator store the escrow key e; having members with an ellipticcurve random number generator send to the administrator, an output rgenerated before an output value of the generator, the administratorlogging the output r for future determination of the state of thegenerator.

BRIEF DESCRIPTION OF THE DRAWINGS

An embodiment of the invention will now be described by way of exampleonly with reference to the appended drawings wherein:

FIG. 1 is a schematic representation of a cryptographic random numbergeneration scheme.

FIG. 2 is a flow chart illustrating a selection process for choosingelliptic curve points.

FIG. 3 is a block diagram, similar to FIG. 1 showing a furtherembodiment

FIG. 4 is flow chart illustrating the process implemented by theapparatus of FIG. 3 .

FIG. 5 is a block diagram showing a further embodiment.

FIG. 6 is a flow chart illustrating yet another embodiment of theprocess of FIG. 2 .

FIG. 7 is schematic representation of an administrated cryptographicrandom number generation scheme.

FIG. 8 is a flow chart illustrating an escrow key selection process.

FIG. 9 is a flow chart illustrating a method for securely utilizing anescrow key.

DETAILED DESCRIPTION OF THE INVENTION

Referring therefore to FIG. 1 , a cryptographic random number generator(ECRNG) 10 includes an arithmetic unit 12 for performing elliptic curvecomputations. The ECRNG also includes a secure register 14 to retain astate value s and has a pair of inputs 16, 18 to receive a pair ofinitialisation points P, Q. The points P, Q are elliptic curve pointsthat are assumed to be known. An output 20 is provided for communicationof the random integer to a cryptographic module 22. The initial contentsof the register 14 are provided by a seed input S.

This input 16 representing the point P is in a first embodiment,selected from a known value published as suitable for such use.

The input 18 is obtained from the output of a one way function in theform of a hash function 24 typically a cryptographically secure hashfunction such as SHA1 or SHA2 that receives as inputs the point P. Thefunction 24 operates upon an arbitrary bit string A to produce a hashedoutput 26. The output 26 is applied to arithmetic unit 12 for furtherprocessing to provide the input Q.

In operation, the ECRNG receives a bit string as a seed, which is storedin the register 14. The seed is maintained secret and is selected tomeet pre-established cryptographic criteria, such as randomness andHamming weight, the criteria being chosen to suit the particularapplication.

In order to ensure that d is not likely to be known (e.g. such thatP=dQ, and ed=1 mod n); one or both of the inputs 16, 18 is chosen so asto be verifiably random. In the embodiment of FIG. 1 , Q is chosen in away that is verifiably random by deriving it from the output of ahash-function 24 (preferably one-way) whose input includes the point P.As shown in FIG. 2 an arbitrary string A is selected at step 202, a hashH of A is computed at step 204 with P and optionally S as inputs to ahash-based function F_(H)( ), and the hash H is then converted by thearithmetic unit 12 to a field element X of a desired field Fat step 206.P may be pre-computed or fixed, or may also be chosen to be a verifiablyrandom chosen value. The field element X is regarded as the x-coordinateof Q (thus a “compressed” representation of Q). The x-coordinate is thentested for validity on the desired elliptic curve E at step 208, andwhether or not X is valid, is determined at step 210. If valid, thex-coordinate provided by element X is decompressed to provide point Q atstep 212. The choice of which of two possible values of the yco-ordinate is generally derived from the hash value.

The points P and Q are applied at respective inputs 16, 18 and thearithmetic unit 12 computes the point sQ where s is the current valuestored in the register 14. The arithmetic unit 12 converts thex-coordinate of the point (in this example point sQ) to an integer andtruncates the value to obtain r=t(z(sQ)). The truncated value r isprovided to the output 20.

The arithmetic unit 12 similarly computes a value to update the register14 by computing sP, where s is the value of the register 14, andconverting the x-coordinate of the point sP to an integer u. The integeru is stored in the register to replace s for the next iteration.

{Ditto Above}

As noted above, the point P may also be verifiably random, but may alsobe an established or fixed value. Therefore, the embodiment of FIG. 1may be applied or retrofitted to systems where certain base points (e.g.P) are already implemented in hardware. Typically, the base point P willbe some already existing base point, such as those recommended inFederal Information Processing Standard (FIPS) 186-2. In such cases, Pis not chosen to be verifiably random.

In general, inclusion of the point P in the input to the hash functionensures that P was determined before Q is determined, by virtue of theone-way property of the hash function and since Q is derived from analready determined P. Because P was determined before Q, it is clearlyunderstood that P could not have been chosen as a multiple of Q (e.g.where P=eQ), and therefore finding d is generally as hard as solving arandom case of the discrete logarithm problem.

Thus, having a seed value S provided and a hash-based function FOprovided, a verifier can determine that Q=F(S,P), where P may or may notbe verifiably random. Similarly, one could compute P=F(S,Q) with thesame effect, though it is presumed that this is not necessary given thatthe value of P in the early drafts of X9.82 were identical to the basepoints specified in FIPS 186-2.

The generation of Q from a bit string as outlined above may be performedexternally of the ECRNG 10, or, preferably, internally using thearithmetic unit 12. Where both P and Q are required to be verifiablyrandom, a second hash function 24 shown in ghosted outline in FIG. 1 isincorporated to generate the coordinate of point P from the bit stringA. By providing a hash function for at least one of the inputs, averifiably random input is obtained.

It will also be noted that the output generated is derived from the xcoordinate of the point sP. Accordingly, the inputs 16, 18 may be the xcoordinates of P and Q and the corresponding values of sP and sQobtained by using Montgomery multiplication techniques thereby obviatingthe need for recovery of the y coordinates.

An alternative method for choosing Q is to choose Q in some canonicalform, such that its bit representation contains some string that wouldbe difficult to produce by generating Q=dP for some known d and P forexample a representation of a name. It will be appreciated thatintermediate forms between this method and the preferred method may alsoexist, where Q is partly canonical and partly derived verifiably atrandom. Such selection of Q, whether verifiably random, canonical, orsome intermediate, can be called verifiable.

Another alternative method for preventing a key escrow attack on theoutput of an ECRNG, shown in FIGS. 3 and 4 is to add a truncationfunction 28 to ECRNG 10 to truncate the ECRNG output to approximatelyhalf the length of a compressed elliptic curve point. Preferably, thisoperation is done in addition to the preferred method of FIGS. 1 and 2 ,however, it will be appreciated that it may be performed as a primarymeasure for preventing a key escrow attack. The benefit of truncation isthat the list of R values associated with a single ECRNG output r istypically infeasible to search. For example, for a 160-bit ellipticcurve group, the number of potential points R in the list is about 2⁸⁰,and searching the list would be about as hard as solving the discretelogarithm problem. The cost of this method is that the ECRNG is madehalf as efficient, because the output length is effectively halved.

Yet another alternative method shown in FIGS. 5 and 6 comprisesfiltering the output of the ECRNG through another one-way functionF_(H2), identified as 34, such as a hash function to generate a newoutput. Again, preferably, this operation is performed in addition tothe preferred method shown in FIG. 2 , however may be performed as aprimary measure to prevent key escrow attacks. The extra hash isrelatively cheap compared to the elliptic curve operations performed inthe arithmetic unit 12, and does not significantly diminish the securityof the ECRNG.

As discussed above, to effectively prevent the existence of escrow keys,a verifiably random Q should be accompanied with either a verifiablyrandom P or a pre-established P. A pre-established P may be a point Pthat has been widely publicized and accepted to have been selectedbefore the notion of the ECRNG 12, which consequently means that P couldnot have been chosen as P=eQ because Q was not created at the time whenP was established.

Whilst the above techniques ensure the security of the system using theECRNG by “closing” the trap door, it is also possible to take advantageof the possible interdependence of P and Q, namely where P=eQ, throughcareful use of the existence of e.

In such a scenario, the value e may be regarded as an escrow key. If Pand Q are established in a security domain controlled by anadministrator, and the entity who generates Q for the domain does sowith knowledge of a (or indirectly via knowledge of d). Theadministrator will have an escrow key for every ECRNG that follows thatstandard.

Escrow keys are known to have advantages in some contexts. They canprovide a backup functionality. If a cryptographic key is lost, thendata encrypted under that key is also lost. However, encryption keys aregenerally the output of random number generators. Therefore, if theECRNG is used to generate the encryption key K, then it may be possiblethat the escrow key e can be used to recover the encryption key K.Escrow keys can provide other functionality, such as for use in awiretap. In this case, trusted law enforcement agents may need todecrypt encrypted traffic of criminals, and to do this they may want tobe able to use an escrow key to recover an encryption key.

FIG. 7 shows a domain 40 having a number of ECRNG's 10 each associatedwith a respective member of the domain 40. The domain 40 communicateswith other domains 40 a, 40 b, 40 c through a network 42, such as theinternet. Each ECRNG of a domain has a pair of identical inputs P,Q. Thedomain 40 includes an administrator 44 who maintains in a secure manneran escrow key e.

) The administrator 44 chooses the values of P and Q such that he knowsan escrow key e such that Q=eP. Other members of the domain 40 use thevalues of P and Q, thereby giving the administrator 44 an escrow key ethat works for all the members of the organization.

This is most useful in its backup functionality for protecting againstthe loss of encryption keys. Escrow keys e could also be mademember-specific so that each member has its own escrow e′ from pointsselected by the administrator 44.

As generally denoted as numeral 400 in FIG. 8 , the administratorinitially selects a point P which will generally be chosen as thestandard generator P for the desired elliptic curve 402. Theadministrator then selects a value d and the point Q will be determinedas Q=dP 404, for some random integer d of appropriate size. The escrowkey e is computed as e=d⁻¹ mod n 406, where n is the order of thegenerator P and stored by the administrator.

The secure use of such an escrow key 34 e is generally denoted bynumeral 500 and illustrated in FIG. 9 . The administrator 44 is firstinstituted 502 and an escrow keys e would be chosen and stored 504 bythe administrator 44

In order for the escrow key to function with full effectiveness, theescrow administrator 44 needs direct access to an ECRNG output value rthat was generated before the ECRNG output value k (i.e. 16) which is tobe recovered. It is not sufficient to have indirect access to r via aone-way function or an encryption algorithm. A formalized way to achievethis is to have each member with an ECRNG 12 communicate with theadministrator 44 as indicated at 46 in FIG. 7 , and step 506 in FIG. 9 .This may be most useful for encrypted file storage systems or encryptedemail accounts. A more seamless method may be applied for cryptographicapplications. For example, in the SSL and TLS protocols, which are usedfor securing web (HTTP) traffic, a client and server perform a handshakein which their first actions are to exchange random values sent in theclear.

Many other protocols exchange such random values, often called nonces.If the escrow administrator observes these nonces, and keeps a log ofthem 508, then later it may be able to determine the necessary r value.This allows the administrator to determine the subsequent state of theECRNG 12 of the client or server 510 (whoever is a member of thedomain), and thereby recover the subsequent ECRNG 12 values. Inparticular, for the client who generally generates a random pre-mastersecret from which is derived the encryption key for the SSL or TLSsession, the escrow key may allow recovery of the session key. Recoveryof the session key allows recovery of the whole SSL or TLS session.

If the session was logged, then it may be recovered. This does notcompromise long-term private keys, just session keys obtained from theoutput of the ECRNG, which should alleviate any concern regardinggeneral suspicions related to escrows.

Whilst escrow keys are also known to have disadvantages in othercontexts, their control within specific security domains may alleviatesome of those concerns. For example, with digital signatures fornon-repudiation, it is crucial that nobody but the signer has thesigning key, otherwise the signer may legitimately argue the repudiationof signatures. The existence of escrow keys means the some other entityhas access to the signing key, which enables signers to argue that theescrow key was used to obtain their signing key and subsequentlygenerate their signatures. However, where the domain is limited to aparticular organisation or part of an organisation it may be sufficientthat the organisation cannot repudiate the signature. Lost signing keysdo not imply lost data, unlike encryption keys, so there is little needto backup signing keys.

Although the invention has been described with reference to certainspecific embodiments, various modifications thereof will be apparent tothose skilled in the art without departing from the spirit and scope ofthe invention as outlined in the claims appended hereto.

1-18. (canceled)
 19. A computer-implemented method of establishing anescrow key for a security domain within a network, comprising:establishing a pair of points (P, Q) as respective inputs to an ellipticcurve random number generator (ECRNG), wherein P=eQ, and e is arelationship between the pair of points (P, Q); generating from theECRNG a random number for use in cryptographic operations within thesecurity domain; using the random number in a cryptographic operation;and logging an output of the ECRNG to reconstruct the random number withthe relationship e as an escrow key.
 20. The computer-implemented methodof claim 19, wherein the pair of points (P, Q) is a pair of ellipticcurve points.
 21. The computer-implemented method of claim 19, whereinone point of the pair of points (P, Q) is obtained from an output of aone way function.
 22. The computer-implemented method of claim 21,wherein the one way function is a hash function.
 23. Thecomputer-implemented method of claim 21, wherein the other point of thepair of points (P, Q) is utilized as an input to the one way function.24. The computer-implemented method of claim 23, wherein the one pointof the pair of points (P, Q) is Q, and the other point of the pair ofpoints (P, Q) is P.
 25. A device, comprising: a memory; and at least oneprocessor communicatively coupled with the memory and configured to:establish a pair of points (P, Q) as respective inputs to an ellipticcurve random number generator (ECRNG), wherein P=eQ, and e is arelationship between the pair of points (P, Q); generate from the ECRNGa random number for use in cryptographic operations within a securitydomain; using the random number in a cryptographic operation; andlogging an output of the ECRNG to reconstruct the random number with therelationship e as an escrow key.
 26. The device of claim 25, wherein thepair of points (P, Q) is a pair of elliptic curve points.
 27. The deviceof claim 25, wherein one point of the pair of points (P, Q) is obtainedfrom an output of a one way function.
 28. The device of claim 27,wherein the one way function is a hash function.
 29. The device of claim27, wherein the other point of the pair of points (P, Q) is utilized asan input to the one way function.
 30. The device of claim 29, whereinthe one point of the pair of points (P, Q) is Q, and the other point ofthe pair of points (P, Q) is P.
 31. A non-transitory computer readablemedium storing instructions which, when executed, cause a computingdevice to perform operations comprising: establishing a pair of points(P, Q) as respective inputs to an elliptic curve random number generator(ECRNG), wherein P=eQ, and e is a relationship between the pair ofpoints (P, Q); generating from the ECRNG a random number for use incryptographic operations within a security domain; using the randomnumber in a cryptographic operation; and logging an output of the ECRNGto reconstruct the random number with the relationship e as an escrowkey.
 32. The non-transitory computer readable medium of claim 31,wherein the pair of points (P, Q) is a pair of elliptic curve points.33. The non-transitory computer readable medium of claim 31, wherein onepoint of the pair of points (P, Q) is obtained from an output of a oneway function.
 34. The non-transitory computer readable medium of claim33, wherein the one way function is a hash function.
 35. Thenon-transitory computer readable medium of claim 33, wherein the otherpoint of the pair of points (P, Q) is utilized as an input to the oneway function.
 36. The non-transitory computer readable medium of claim35, wherein the one point of the pair of points (P, Q) is Q, and theother point of the pair of points (P, Q) is P.